6-2 Properties of Parallelograms
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1 6-2 Properties of Parallelograms Warm Up Lesson Presentation Lesson Quiz Geometry
2 Warm Up Find the value of each variable. 1. x 2. y 3. z
3 Objectives Prove and apply properties of parallelograms. Use properties of parallelograms to solve problems.
4 parallelogram Vocabulary
5 Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals are given their own names.
6 Helpful Hint Opposite sides of a quadrilateral do not share a vertex. Opposite angles do not share a side.
7 A quadrilateral with two pairs of parallel sides is a parallelogram. To write the name of a parallelogram, you use the symbol.
8
9
10 Example 1A: Properties of Parallelograms In CDEF, DE = 74 mm, DG = 31 mm, and m FCD = 42. Find CF. opp. sides CF = DE CF = 74 mm Def. of segs. Substitute 74 for DE.
11 Example 1B: Properties of Parallelograms In CDEF, DE = 74 mm, DG = 31 mm, and m FCD = 42. Find m EFC. m EFC + m FCD = 180 m EFC + 42 = 180 m EFC = 138 cons. s supp. Substitute 42 for m FCD. Subtract 42 from both sides.
12 Example 1C: Properties of Parallelograms In CDEF, DE = 74 mm, DG = 31 mm, and m FCD = 42. Find DF. DF = 2DG DF = 2(31) DF = 62 diags. bisect each other. Substitute 31 for DG. Simplify.
13 Check It Out! Example 1a In KLMN, LM = 28 in., LN = 26 in., and m LKN = 74. Find KN. opp. sides LM = KN LM = 28 in. Def. of segs. Substitute 28 for DE.
14 Check It Out! Example 1b In KLMN, LM = 28 in., LN = 26 in., and m LKN = 74. Find m NML. NML LKN m NML = m LKN m NML = 74 opp. s Def. of s. Substitute 74 for m LKN. Def. of angles.
15 Check It Out! Example 1c In KLMN, LM = 28 in., LN = 26 in., and m LKN = 74. Find LO. LN = 2LO 26 = 2LO LO = 13 in. diags. bisect each other. Substitute 26 for LN. Simplify.
16 Example 2A: Using Properties of Parallelograms to Find Measures WXYZ is a parallelogram. Find YZ. opp. s YZ = XW 8a 4 = 6a a = 14 a = 7 Def. of segs. Substitute the given values. Subtract 6a from both sides and add 4 to both sides. Divide both sides by 2. YZ = 8a 4 = 8(7) 4 = 52
17 Example 2B: Using Properties of Parallelograms to Find Measures WXYZ is a parallelogram. Find m Z. m Z + m W = 180 cons. s supp. (9b + 2) + (18b 11) = 180 Substitute the given values. 27b 9 = 180 Combine like terms. 27b = 189 Add 9 to both sides. b = 7 Divide by 27. m Z = (9b + 2) = [9(7) + 2] = 65
18 Check It Out! Example 2a EFGH is a parallelogram. Find JG. diags. bisect each other. EJ = JG Def. of segs. 3w = w + 8 Substitute. 2w = 8 Simplify. w = 4 Divide both sides by 2. JG = w + 8 = = 12
19 Check It Out! Example 2b EFGH is a parallelogram. Find FH. diags. bisect each other. FJ = JH 4z 9 = 2z 2z = 9 Def. of segs. Substitute. Simplify. z = 4.5 Divide both sides by 2. FH = (4z 9) + (2z) = 4(4.5) 9 + 2(4.5) = 18
20 Remember! When you are drawing a figure in the coordinate plane, the name ABCD gives the order of the vertices.
21 Example 3: Parallelograms in the Coordinate Plane Three vertices of JKLM are J(3, 8), K( 2, 2), and L(2, 6). Find the coordinates of vertex M. Since JKLM is a parallelogram, both pairs of opposite sides must be parallel. Step 1 Graph the given points. K L J
22 Example 3 Continued Step 2 Find the slope of by counting the units from K to L. The rise from 2 to 6 is 4. The run of 2 to 2 is 4. L K Step 3 Start at J and count the same number of units. A rise of 4 from 8 is 4. A run of 4 from 3 is 7. Label (7, 4) as vertex M. J M
23 Example 3 Continued Step 4 Use the slope formula to verify that K L J M The coordinates of vertex M are (7, 4).
24 Check It Out! Example 3 Three vertices of PQRS are P( 3, 2), Q( 1, 4), and S(5, 0). Find the coordinates of vertex R. Since PQRS is a parallelogram, both pairs of opposite sides must be parallel. Step 1 Graph the given points. P Q S
25 Check It Out! Example 3 Continued Step 2 Find the slope of from P to Q. The rise from 2 to 4 is 6. The run of 3 to 1 is 2. Step 3 Start at S and count the same number of units. A rise of 6 from 0 is 6. by counting the units A run of 2 from 5 is 7. Label (7, 6) as vertex R. P Q S R
26 Check It Out! Example 3 Continued Step 4 Use the slope formula to verify that P Q S R The coordinates of vertex R are (7, 6).
27 Example 4A: Using Properties of Parallelograms in a Proof Write a two-column proof. Given: ABCD is a parallelogram. Prove: AEB CED
28 Proof: Example 4A Continued Statements Reasons 1. ABCD is a parallelogram 1. Given 2. opp. sides 3. diags. bisect each other 4. SSS Steps 2, 3
29 Example 4B: Using Properties of Parallelograms in a Proof Write a two-column proof. Given: GHJN and JKLM are parallelograms. H and M are collinear. N and K are collinear. Prove: H M
30 Proof: Example 4B Continued Statements 1. GHJN and JKLM are parallelograms. 2. H and HJN are supp. M and MJK are supp. 3. HJN MJK Reasons 1. Given 2. cons. s supp. 3. Vert. s Thm. 4. H M 4. Supps. Thm.
31 Check It Out! Example 4 Write a two-column proof. Given: GHJN and JKLM are parallelograms. H and M are collinear. N and K are collinear. Prove: N K
32 Proof: Check It Out! Example 4 Continued Statements Reasons 1. GHJN and JKLM are parallelograms. 2. N and HJN are supp. K and MJK are supp. 3. HJN MJK 4. N K 1. Given 2. cons. s supp. 3. Vert. s Thm. 4. Supps. Thm.
33 Lesson Quiz: Part I In PNWL, NW = 12, PM = 9, and m WLP = 144. Find each measure. 1. PW 2. m PNW
34 Lesson Quiz: Part II QRST is a parallelogram. Find each measure. 2. TQ 3. m T 28 71
35 Lesson Quiz: Part III 5. Three vertices of ABCD are A (2, 6), B ( 1, 2), and C(5, 3). Find the coordinates of vertex D. (8, 5)
36 Lesson Quiz: Part IV 6. Write a two-column proof. Given: RSTU is a parallelogram. Prove: RSU TUS Statements Reasons 1. RSTU is a parallelogram. 1. Given 2. cons. s 3. R T 4. RSU TUS 3. opp. s 4. SAS
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